rank tests of unit root hypothesis with infinite variance errors

Journal of Econometrics 104 (2001) 49–65www.elsevier.com/locate/econbase

Rank tests of unit root hypothesis with in$nitevariance errors

Mohammad N. Hasana ;∗aAmerican Express/TRS/CIM, APCC Bldg., 7740 N 16th Street, Phoenix,

AZ 85020, USA

Received 25 April 1997; revised 15 October 2000; accepted 22 January 2001

Abstract

We consider a family of rank tests based on the regression rank score processintroduced by Gutenbrunner and Jure0ckov1a (Ann. Statist. 20 (1992) 305) to test theunit root hypothesis under in$nite variance innovations. Unlike the $nite variancecase as studied by Hasan and Koenker (Econometrica 65 (1997) 133) the originalrankscore test statistics (Tn) exhibit a simple Gaussian limiting behavior. However,$nite sample investigations suggest a correction similar to what HK proposed. Thiscorrected version (ST ) has reliable size, and exhibits remarkable power even in nearunit root cases under a variety of �-stable distributions. Also, the test statistics donot depend on the � parameter. ? 2001 Elsevier Science S.A. All rights reserved.

JEL classi,cation: C12; C22

Keywords: Unit root tests; �-stable process; Regression rank score; Quantile regression

1. Introduction

Although the “in$nite variance paradigm” has never quite captured a cen-tral place in the empirical modeling of time-series data, on theoretical front,in both statistics and econometrics alike there has been growing interest inthis problem. The statistical modeling of in$nite variance error is tradition-ally couched in the framework of the domain of attraction of a stable law.The same approach will be pursued here. Mikosch et al. (1995) considered

∗ Corresponding author. Tel.: +1-602-766-9273; fax: +1-602-766-9207.E-mail address: [email protected] (M.N. Hasan).

0304-4076/01/$ - see front matter ? 2001 Elsevier Science S.A. All rights reserved.PII: S0304-4076(01)00050-1

50 M.N. Hasan / Journal of Econometrics 104 (2001) 49–65

the estimation of a ARMA process with �-stable innovations. They exhib-ited estimators which are consistent, and converge to the true values fasterthan those in the usual case with $nite variance innovations. KlGuppelbergand Mikosch (1996) studied the asymptotic behavior of the integrated peri-odogram for �-stable process and applied their results to explore some speci$cgoodness-of-$t tests for heavy-tailed linear processes. In a recent survey pa-per, McCulloch (1997) provides some empirical support for the stable modelfor $nancial data.More recently, this interest of in$nite variance has also permeated the unit

root literature of time-series econometrics. The classical asymptotic results,like the $nite variance case are non-standard; least squares (LS) based anal-ysis of unit root with in$nite variance error was studied by Chan and Tran(1989). Phillips (1990) showed that the limiting random variables are ratiosof stochastic integrals involving Wiener and L1evy processes. Furthermore,the convergence rate of the LS estimators for the in$nite variance case re-mains n−1 as under the $nite variance case, failing to capture the potentialadvantage of the “natural experiment” of the design matrix.It is well documented in innovation-outlier time-series models (Martin,

1980, 1981) that many robust estimators perform better than their LS counter-parts in statistical inference by taking advantage of the good leverage pointsand down-weighting bad leverage points as much as possible. This feature wasalso evidenced in the recent M -estimation analysis of the integrated in$nitevariance process by Knight (1989, 1991). He shows that certain M -estimatesincluding LAD converge faster than LS estimators and that they are alsoasymptotically normal which separates them further from LS based analysis.Thus, if the innovations happen to come from Cauchy distribution (� = 1),the rate of convergence will be n−3=2, a factor of n−1=2 improvement over therate of convergence of LS estimators. The faster convergence yields improvedpower of the test statistics for the unit root hypothesis.Regression rank scores based testing procedures, recently developed by

Gutenbrunner et al. (1993) (hereafter GJKP) for the linear regression model,share exactly the same advantages as M -estimation while avoiding somecollateral problems of estimating the nuisance parameters. This fact distin-guishes them from direct Wald, likelihood ratio type tests based on robustM -estimators, as for example, the LAD tests of Knight (1991).The virtue of classical rank tests under the simple null hypothesis includ-

ing distribution free, exact, easy to compute and robustness behavior hasbeen appealing to many time-series econometricians. In their pioneering ef-forts, Dufour (1981), Campbell and Dufour (1995, 1997) have consideredseveral variants of sign and signed rank tests designed to test orthogonal-ity restrictions including random walk hypothesis. Extending the rank testsfrom simple null to composite null within a general context of time serieseconometrics poses a severe challenge. It has, however, been attempted by

M.N. Hasan / Journal of Econometrics 104 (2001) 49–65 51

preliminary estimation yielding aligned rank tests, although in most of thecases at the expense of obtaining only an asymptotically justi$ed procedure.Adichie (1986), Puri and Sen (1985), Campbell and Dufour (1997), andHallin and Puri (1992, 1994) are some examples of such testing procedure.The history of rank tests has been long and illustrious. We refer interestedreaders to a recent survey paper by Koenker (1996).An alternative approach based on regression ranks scores developed by

GJKP does not require alignment or any preliminary estimation to generatingthe ranks of the observations. Although a full treatment of ARMA models hasyet to develop, Koul and Saleh (1995) provided a rather detailed treatmentof AR model without explicitly addressing the testing problems. Hasan andKoenker (1997) (hereafter HK) have employed the GJKP approach for unitroot model with $nite variance error structure. The purpose of this paper isto extend the work of the latter authors to the in$nite variance case.

2. Preliminary results

The data generation process considered in this paper is

yt = yt−1 + ut; t = 1; : : : ; n; (2.1)

ut = �1ut−1 + · · ·+ �put−p + et ; (2.2)

where the roots of 1 − �1z − · · · − �pzp = 0 are all assumed to lie outsidethe unit circle. Our tests will be based on the augmented Dickey and Fuller(1979) model

Nyt = �0 + �yt−1 +p∑

j=1�jNyt−j + et ; (2.3)

where we wish to test the hypothesis, H0: � = 0. Partitioning the designmatrix, we will write xt = (1; Nyt−1; : : : ;Nyt−p)′; zt = yt−1, and in matrixnotation z=X (X ′X )−1X ′z. Following Koenker and Bassett (1978) a sequenceof (p+ 1)-dimensional “regression quantiles” for the above null model maybe de$ned as

�n(�) = argminb∈R(p+1)

n∑i=1

��(Nyt − x′ib) (2.4)

with ��(e)= e(�− I(e¡ 0)), which may be regarded as estimators of the co-ePcients of the (p+ 1)-dimensional conditional quantile functions of Ny|x.The regression rank score process as de$ned by Gutenbrunner and Jure0ckov1a(1992) can be described by reformulating the problem (2.4) as a linear pro-gram with corresponding dual problem

a(�) = argmax{Ny′a|X′a= (1− �)X′1n; a ∈ [0; 1]n}: (2.5)


Table 1ARE of Wilcoxon vs. F-tests

F Normal Uniform Logistic DExp t3 Exp LNormal t2

ARE 0.955 1.0 1.097 1.5 1.9 3 7.35 ∞

In the simple one-sample location problem, where X=1n, the regression rankscore process for the tth observation, at(�) specializes to the rank generatingfunctions of H1ajek and 0Sid1ak (1967). For linear regression model with nui-sance parameters an(�) are more complex but provide a natural extension ofranking the observations. One very important feature is that an(�) is piecewiselinear and continuous with breaks at 0= �0 ¡�1 · · ·¡�J−1 = 1. Koenker andd’Orey (1987, 1993) have considered the problem of computing the entiresample path of an(�). This piecewise linearity of an(�) can be exploited incomputing the integrals

bn = (bnt)nt=1 =

(∫ 1

0at(�) d’(�)

)n

t=1

=

(−∫ 1

0’(�) dat(�)

)n

t=1

(2.6)

which play an important role to construct the rank based tests to be investi-gated in this paper, as the following examples illustrate.Wilcoxon scores: ’(s) = s− 1=2: Clearly T’= E’(s) =

∫ 10 ’(s) ds= 0, and

A2(’) = E’2(s) =∫ 10 ’2(s) ds= 1=12. Integration by parts yields

bnt=−∫ 1

0(s−1=2) dat(s)=

J∑j=1

1=2(at(�j+1)+at(�j)) (�j+1−�j)−1=2:

Wilcoxon scores are asymptotically optimal when the underlying density, fof the error term is logistic. Hodges and Lehmann (1956) established a lowerbound to the ARE of Wilcoxon scores relative to t-test about 0.864 forany distribution, F , having an L2 norm of the density function. Thus, innon-Gaussian cases, if one naively uses t-test, it can never be more thanabout 15% better than the Wilcoxon test of location shift, but can be arbi-trarily worse as indicated in Table 1.Normal (van der Waerden) scores: ’(s)=#−1(s); 0¡s¡ 1. It is evident

that T’ = 0; A2(’) = 1. Denoting the standard normal df and density by #and � respectively, and using the fact that the function a′i(�) is piecewiseconstant,

bnt =−∫ 1

0#−1(s) dat(s) =

J∑j=1

a′t(�j)[�(#−1(�j+1))− �(#−1(�j))]:

These normal scores are optimal when f is Gaussian. ChernoV and Savage(1958) showed that ARE of rank tests of location shift relative to F testexceeds unity for all distribution functions, F , except the normal distributionwhere ARE = 1.


Sign-median scores: ’(s) = 12sgn(s− 1

2 ). Again T’= 0; A2(’) = 1=4, and

bnt =−∫ 1

0’(s) dat(s) = at(1=2)− 1=2:

They are optimal for f double exponential. The above optimality results areknown only for $nite variance errors. For the in$nite variance case we wouldlike to illuminate this issue by Monte Carlo experiments in Section 4.For the $nite variance case HK (1997) proposed a modi$cation of GJKP

test statistics based on the regression rank score process ant(�) for the unitroot null model (2.3) with �=0. The modi$cation required consideration ofthe bivariate process {(ut; vt)}, where vt = ’(F(et)), for some appropriatelychosen score function, such that Evt = 0, and Var(vt) = A2(’). Now de$ne&= limT→∞Var(T−1=2∑(ut; vt)′) and consider the lower triangular form ofthe Cholesky decomposition of & as

&1=2 = &−1=211

[&11 0

&12 '1=2

]; (2.7)

where '= |&|. The original GJKP rank test statistics, ST , of unit root modelhave the following asymptotic distribution:

('1=2 ST ⇒ &12

'1=2

∫ 10 (W1 − TW 1) dW1

(∫ 10 (W1 − TW 1)2 ds)1=2

+

∫ 10 (W1 − TW 1) dW2

(∫ 10 (W1 − TW 1)2 ds)1=2

:

To get back the Gaussian limiting behavior of the test statistics, followingPhillips (1987), HK (1997) considered the test statistics of the form

ST =(

'1=2 ST − &12(z

'1=2(

[T∑

(zt − Tz)Nyt∑(zt − zt)2

− ((2 − (2u)

2(2z

]; (2.8)

where

ST =[∑

(zt − zt)2)]−1=2∑

(zt − zt)bnt ; bnt =∫ 1

0ant(�) d’(�);

�T =[∑

(zt − zt)2]−1∑

(zt − zt)Nyt;

(2 = &11 = limT→∞

1TE(∑

ut)2 ; (2

u = limT→∞

1T∑Eu2t ;

(2z =

1T 2

∑(zt − zt)2:

Replacing Tz=∑

yt−1=T by zt=X (X ′X )−1X ′zt in the numerator of the secondterm, HK (1997) reported the following asymptotically equivalent version of(2.8):

ST =(

'1=2 ST − &12(z

'1=2(

[T�T − (2 − (2

u

2(2z

]: (2.9)


However, in the in$nite variance case, as stated in Theorem 3.4, the originalversion of the GJKP test statistics have the following limiting distribution:

Tn = A−1(’)ST

=∑

(zt − zt)bnt

[A2(’)∑

(zt − zt)2]1=2

⇒∫ 10 (S(s)− TS) dW (s)

[∫ 10 (S(s)− TS)2]1=2

: (2.10)

In Section 3, under the Assumptions of (A1)–(A4), Theorem 3.4 and Corol-lary 3.5 we show that the test statistics Tn asymptotically follow Gaussiandistribution. In addition, the test statistics do not depend on the � para-meter which characterizes the tail behavior of the �-stable distribution. Thisis rather surprising and very attractive feature of the test statistics since theePcient estimation of the � parameter from the observed data is a quite dif-$cult problem. For details see DuMouchel (1983), Hill (1975) and a recentpaper by McCulloch (1997).

2.1. Finite sample correction

The $ndings in Tables 2 and 3 suggest some $nite sample correction bemade to improve the size of the test statistics, Tn. Fortunately, the similartype of correction factor as suggested by HK (1997) for the $nite variancecase works here as well. It is instructive to understand why. The key fac-tor is the asymptotic distribution of

∑(zt − zt)bnt—the numerator of ST and

Tn. While in the $nite variance case, its distribution, as in (3.9), is derivedfrom the bivariate principle, the analogue in the in$nite variance is givenby Lemma 3.3, which is a simpler expression because of the asymptotic in-dependence of

n−1=2[ns]∑t=1

’(F(et)) and c−1n

[ns]∑t=1

et : (2.11)

In the $nite variance case, the covariance between these two processes iscaptured by a non-zero value of the &12 term. Substituting &12 = 0, and con-sequently '1=2 = (A(’) in (2.8) or (2.9), we would get the test statistics,(2.10). Thus, the test statistics Tn; ST , and ST are asymptotically equivalentunder the in$nite variance case. The ,nite sample dependence of the bivariateprocesses in (2.11) is conjectured to cause the size distortion of these sim-pli$ed test statistics, Tn, as reported in Table 2. Further reYection suggeststhat it is rather natural to have some $nite sample dependence between thesetwo processes, since they are essentially generated by the same innovations.


Table 2Empirical size of the test statistics (Tn)a

Index of stable Score function Sample size ndistribution

100 200 400

� = 1:2 Wilcoxon 0.225 0.190 0.159Normal 0.273 0.232 0.208Sign median 0.145 0.130 0.108

�= 1 Wilcoxon 0.181 0.143 0.117(Cauchy) Normal 0.232 0.187 0.158

Sign median 0.121 0.099 0.087

�= 0:8 Wilcoxon 0.138 0.113 0.102Normal 0.183 0.150 0.135Sign median 0.096 0.084 0.083

�= 0:5 Wilcoxon 0.114 0.127 0.140Normal 0.148 0.156 0.150Sign median 0.094 0.120 0.163

aNumber of replications: 5000, nominal signi$cance of the test is 5%, null model: Nyt=++et ,where et is i.i.d. symmetric stable random variable.

Table 3Empirical power and size of the test statistics (Tn)a

Index of Score function Power without size correction Size Size corrected powerstable distribution

0.90 0.95 0.99 1.00 0.9 0.95 0.99

�= 1:2 Wilcoxon 1.00 0.986 0.669 0.225 0.993 0.918 0.371Normal 1.00 0.985 0.680 0.273 0.979 0.857 0.309Sign median 0.998 0.968 0.582 0.145 0.983 0.908 0.392

�= 1 Wilcoxon 1.00 0.998 0.831 0.181 0.999 0.988 0.648(Cauchy) Normal 1.00 0.998 0.824 0.232 0.998 0.971 0.574

Sign median 0.999 0.995 0.799 0.121 0.996 0.986 0.694



aSample size: 100, number of replications: 5000, nominal signi$cance of the test is 5%.There are three stationary alternatives (0.9, 0.95, 0.99), null model: Nyt = + + et , where et isi.i.d. symmetric stable distribution with index parameter, �.


We propose ST in (2.9) as feasible in$nite variance test statistics which areasymptotically equivalent to the $nite variance test statistics in (2.8).In what follows we make the following assumptions about the sequence

of {et}.A1: {et} is an i.i.d sequence with E(e2t ) =∞, and et belongs to a domain of

attraction of stable law with index � ∈ (0; 2).A2: The partial sum process Sn(·) = c−1

n∑[n·]

t=1et converges in distribution inD[0; 1] (the space of right-continuous functions with left limits) to aprocess S(·). The normalizing constants {cn} satisfy, for � ∈ (0; 2),

nP[|et |¿cnx] → x−� or for �= 2; nc−2n E{e2t I(|et |6 cn)} → 1:

It can also be shown that cn = n1=�‘(n) where ‘(·) is a slowly varyingfunction.

A3: F has an absolutely continuous density f, and $nite Fisher Information.A4: ’ : [0; 1] → R is a nondecreasing square integrable function on the in-

terval [�0; 1 − �0] for some �0 ∈ (0; 0:5) and ’ is constant on the set[0; �0] ∪ [1− �0; 1].

Assumption A1 implies that there exist centering constants bn such thatSn(·) − [n · ]bn converges in distribution to a process S(·) (see Resnick andGreenwood, 1979). This is essentially the Central Limit Theorem for in$-nite variance case and thus excludes the Gaussian distribution case of �= 2.Assumption A2 is just a re$nement of A1. Assumptions A3 and A4 arerequired to study the asymptotic behavior of regression rank score processbased on which we construct our test statistics, but imposes very weak re-strictions on the distribution of the innovation process. Indeed, the assumptionof constancy of ’ function in the tails of [0; 1] in A4 limits the choice of ascore function but makes the asymptotics considerably simpler. Like the $nitevariance case as shown by HK (1997) our Monte Carlo experiments showthat score functions like Wilcoxon and normal which strictly do not satisfyA4, still perform very satisfactorily in terms of their size and power. This isprobably because the contributions to the regression rank score statistics fromthe extreme tails are inherently limited by the boundedness of the regressionrank score functions.We shall now give some results concerning the joint limiting behavior of

partial-sum processes de$ned in terms of random variables {et} satisfyingassumptions A1 and A2. Consider the following two empirical processes:

Wn(s) = n−1=2[ns]∑t=1

’(F(et)) and Sn(s) = c−1n

[ns]∑t=1

et : (2.12)

It is well known that Wn(·) ⇒ W (·), a standard Brownian-motion process.Resnick and Greenwood (1979) showed that if Sn(·) ⇒ S(·) (by A2, in


this case) then(Sn(·)Wn(·)

)⇒(

S(·)W (·)

)on D[0; 1]×D[0; 1]; (2.13)

where S(·) and W (·) are independent. This asymptotic independence of the bi-variate process, (Sn;Wn) has an important role to play in determiningthe asymptotic properties of linear regression rank score statistics.

3. Asymptotics

Consider the ADF model (2.3), Nyt = �0 + �yt−1 +∑p

j=1�jNyt−j + et ,and let

xt = (1;Nyt−1; : : : ;Nyt−p; n−1=2yt−1)′: (3.1)

Note that, Nyt is a stable AR(p) process and hence

max16t6n

|Nyt |=Op(cn) and max16t6n

||xt ||=Op(cn) (3.2)

with cn as de$ned in A2, where cn=√n → ∞. Furthermore,

dn = c−2n∑xtx

′t =Op(1): (3.3)

The following lemma which adapts Lemma A:1 of HK (1997) to the presentsetting provides a quadratic approximation of the quantile regression objectivefunction uniformly in � and the unknown parameters of the model. It isessential for the asymptotic linear representation of the proposed test statistics(2.10).

Lemma 3.1. Under the conditions in A1–A3; for any ,xed �0 ∈ (0; 0:5); andC ¿ 0;

sup{|Rn(b; �)| : ||b||¡C; �0 6 �6 1− �0} →p 0 (3.4)

as n → ∞; where

Rn(b; �) = 4(�)(−1�

n∑t=1

[��(et� − c−1n (�x

′tb)− ��(et�)]

+c−1n 4(�)

n∑t=1x′tb �(et�)− 1

2b′dnb;

et� = et−F−1(�); 4(�)= (�(1−�))−1=2; �(x)= �−I [x¡0]; and (� = 4(�)=f(F−1(�)).

Proof. Follows exactly the same argument as in Lemma A:1 of HK (1997),with the obvious modi$cation in the normalizing constants appeared in Lem-mas 3.2 and 3.3. The rationale for these normalizing constants and the de$ni-tion of xt is that the $rst p components of the estimator of b are cn-consistent,


i.e. cn(�jn − �j) = Op(1), and the last component is cn√n-consistent, i.e.

cn√n�n=Op(1). Thus, employing a truncation method $rst, for S ≡ {(b; �): �0

6 �6 1− �0; ||b||6 C}, we show

supS|Rn(b; �)|I [||c−1

n x′tb||6 B] →p 0: (3.5)

Then we evaluate the approximation error due to the truncation. A similartechnique has been used by Davis et al. (1992) and Knight (1989, 1991).Now recalling the notation in Section 2, let zt=zt− zt , where z=X (X ′X )−1

X ′z. The following lemma approximates the regression rank score process byan empirical process using the arguments for Lemmas A:2 and A:3 in HK(1997).

Lemma 3.2. Assume conditions of Lemma 3.1; and let at(�) = I(et ¿F−1(t)); for t = 1; : : : ; n; then as n → ∞;

sup�06�61−�0

{∣∣∣∣c−1n n−1=24(�)

n∑t=1

zt(at(�)− at(�))∣∣∣∣}

→p 0: (3.6)

Remark. Note that the conditions of Lemma A:2 in HK (1997) follow im-mediately in the present circumstances since

∑x′t zt = 0 and c−2

n n−1∑z2t =(2z = Op(1). In fact, using Lemma 2 of Knight (1991) one can easily verify

that

c−2n n−1∑z2t ⇒ !2

∫ 1

0(S(s)− TS)2 ds; (3.7)

where TS =∫ 10 S(s) ds, and !=

∑∞0 dj, with ut =

∑∞j=0djet−j. In our present

setting the coePcients (dj) of the moving average representation of a stableAR(p) process of ut easily satisfy the assumption A3 in Knight (1991).Our next objective is to obtain an asymptotic representation for linear rank

score statistics c−1n n−1=2∑n

t=1zt bnt , where bnt is de$ned in (2.6). Here weneed to apply the condition A4 in Section 2. Again, following the argumentin Lemmas A:4 and A:5 of HK (1997) and using the appropriate normalizingconstants for the in$nite variance case we can establish a linear representationof the following form:

c−1n n−1=2

n∑t=1

zt bnt = c−1n n−1=2

n∑t=1

zt’(F(et)) + op(1): (3.8)

We are now interested in the limiting distribution of c−1n n−1=2∑n

t=1zt’(F(et)).De$ning vt = ’(F(et)), for $nite variance case, HK (1997) in Lemma A:6made use of a bivariate invariance principle to show that

n−1n∑

t=1ztvt ⇒ &12

∫ 1

0W1(s) dW1(s) +N1=2

∫ 1

0W1(s) dW2(s); (3.9)


where W1 and W2 are the two elements of the bivariate Brownian motion.Note that &12 represents the covariance of the bivariate process which takesnon-zero value in the $nite variance case. However, as indicated in Section2, this term vanishes in the in$nite variance case because of the asymptoticindependence of the bivariate processes n−1=2∑[ns]

t=1’(F(et)) and c−1n∑[ns]

t=1et .

Lemma 3.3. Under the conditions in A1–A4; as n → ∞

c−1n n−1=2

n∑t=1

zt’(F(et)) ⇒ A(’)!∫ 1

0(S(s)− TS) dW (s) (3.10)

where A2(’) =∫ 10 ’2(s) ds; zt = zt − zt ; and ! is de,ned as in (3:7).

Proof. The proof is immediate using Lemma 2 of Knight (1991).

3.1. Distribution of the test statistics under null

Recall the augmented Dickey–Fuller model in (2.3), and the test statisticTn in (2.10). The following theorem provides the asymptotic distribution ofTn under H0.

Theorem 3.4. Assume that conditions (A1)–(A4) hold. Then as n → ∞

Tn =∑

(zt − zt)bnt

[A2(’)∑

(zt − zt)2]1=2 ⇒

∫ 10 (S(s)− TS) dW (s)

[∫ 10 (S(s)− TS)2]1=2

: (3.11)

Proof. Observe that by asymptotic linear representation of regression rankscore statistics in (3.8) and by Lemma 3.3, we have

[A2(’)]−1=2c−1n n−1=2

n∑t=1

(zt − zt)bnt ⇒ !∫ 1

0(S(s)− TS) dW (s):

Also, from (3.7) we can deduce that

c−1n n−1=2 [∑(zt − zt)2

]1=2 ⇒ !

[∫ 1

0(S(s)− TS)2 ds

]1=2:

Therefore, by the continuous mapping theorem, the expression in (3.11) fol-lows.

Corollary 3.5. Under the same conditions as in Theorem 3.4; and condi-tional on (S(s): 0 6 s 6 1); the test statistic Tn asymptotically follows anormal distribution with mean zero and variance one.

Since S(s) and W (s) are independent stochastic processes, therefore, con-ditional on (S(s): 06 s6 1), the numerator of the right-hand side of (3.11)


has normal distribution with variance∫ 10 (S(s)− TS)2 ds implying Tn ⇒ W(1) ≡

N(0; 1).

4. Monte-Carlo experiment

To ensure two signi$cant digits of accuracy, 5000 replications were donethroughout. For each replication, an initial 125 observations were generatedand then the $rst 25 are discarded to take into account the start-up eVects.Thus, our eVective sample size is 100 for most of the experiments. For someexperiments as in Table 2 initially 250 and 500 observations are generated toget eVective sample size of 200 and 400, respectively. The stable distributionsare generated in Splus, using the method of Chambers et al. (1976). EPcientsimulation of stable random variables is an interesting open problem, see, forexample Janicki and Weron (1994). The Monte-Carlo experiment is carriedout in Splus using algorithms for the underlying test statistics written inFortran, and which are available from the author on request.To study the $nite sample performance of the tests, Tn we begin with

generating the data according to the following law:

yt = yt−1 + et ; (4.1)

where et is i.i.d. and belongs to a family of symmetric stable distributionwhich is characterized by the index �. Normal distribution corresponds to�=2, whereas �=1 represents Cauchy distribution. The index � controls thethickness of the tail of the distribution of stable random variable. The smallerthe value of � the heavier the tail of the distribution. In this experiment wechoose $ve diVerent values of � to see how the distributional diVerencesaVect the estimated size and power of the test statistics. These values are2; 1:2; 1; 0:8, and 0.5.Table 2 reveals the slow convergence of the test-statistics (2.10) to their

asymptotic normal distribution. For 5% nominal level of signi$cance the clos-est estimated size for a sample of size 100 is 9:4% for a stable random vari-able with index parameter �= 0:5. However, for the � value in the range of(1:2; 1 and 0.8) the estimated size gradually improves and converges towardsits nominal 5% level of signi$cance as we increase the sample size from 100to 200 and $nally to 400.Because of the size distortion we have computed an empirically size cor-

rected power of the test statistics (2.10). As Table 3 indicates the distributionswith thicker tails have better power performance, as the stationary parameterconverge to unit root. For example, while the size corrected power againstthe 0.99 stationary alternative for stable random variable with �= 1:2 are inthe vicinity of 35% the corresponding $gures jump to as high as 69% and100% for �= 1 and 0.5, respectively.


4.1. Performance of the feasible test statistics

The $ndings in Tables 2 and 3 suggest to consider two asymptoticallyequivalent version of feasible test statistics, as discussed in Section 2. Thetest statistics, ST , in (2.8) for the $nite variance case are called the Zbarversion, and ST in (2.9) for the in$nite variance case are called the Zhatversion. To investigate the $nite sample performance of these feasible teststatistics we now consider a somewhat more general process

yt = yt−1 + ut; ut = 0:5ut−1 + et ;

where et is chosen from the family of stable distributions with �∈(2; 1:2; 1;0:8; 0:5).To construct the feasible test statistics we compute the following null

model:

Nyt = ++ �1Nyt−1 + etby the l1-$t criteria, and obtain u t = Nyt − e t , vt = ’(F(e t)), and F(e t) =(#{e s 6 e t ; s = 1; : : : ; T} − 0:5)=T . We may now estimate the nuisanceparameters as

&11 = (2 = T−1T∑

t=1u2t + 2T−1

l∑j=1

w(j; l)T∑

t=1u t u t−j;

&12 = T−1T∑

t=1u t vt + T−1

l∑j=1

w(j; l)T∑

t=1(u t vt−j + vt u t−j);

&22 = T−1T∑

t=1v2t + 2T−1

l∑j=1

w(j; l)T∑

t=1vt vt−j;

where we chose to use !(j; l)=1− j=(1+ l), and the integer l is referred toas the lag truncation parameter. Finally, we estimate (2

u by (2u=T−1∑u2t and

the determinant of the variance-covariance matrix (&) by N=&11&22−(&12)2.For further details the interested reader may refer to HK (1997).Tables 4 and 5 report the empirical size and power of the rank test statistics.

The estimated size of the rank tests appears to be very close to nominal sizeof the tests. “Zhat” version (ST ) in Table 5, yields better size with similarpower than their counterparts of “Zbar” version (ST ) in Table 4, for allvalues of �¡ 2. The lag truncation parameter, l, does not seem to have anysigni$cant eVect on power and size of the tests, as we increase it from l=2to 6. Power of the rank tests continued to be spectacular for a wide varietyof stable distributions with in$nite variance errors. With regard to the choiceof a particular score function, all three have very satisfactory performance.Although one may be led to prefer the Wilcoxon or normal scores tests onthe basis of optimality considerations, it should be emphasized that only thesign score based test is consistent with the assumption A4 in Section 2.


Table 4Finite sample correction: Zbar version (ST )a

Index of Test statistic l= 2 l= 6stable distribution

Power Size Power Size0.90 0.95 0.99 1.00 0.90 0.95 0.99 1.00

�= 2 ADF 0.276 0.125 0.061 0.051 0.276 0.125 0.061 0.051(normal) Wilcoxon 0.162 0.128 0.085 0.065 0.145 0.116 0.080 0.066

Normal 0.173 0.134 0.082 0.070 0.160 0.128 0.083 0.072Sign 0.127 0.123 0.094 0.077 0.114 0.107 0.091 0.076

ADF 0.144 0.071 0.039 0.050 0.144 0.071 0.039 0.050�= 1:2 Wilcoxon 0.929 0.875 0.461 0.093 0.874 0.827 0.436 0.095

Normal 0.893 0.837 0.429 0.096 0.837 0.782 0.400 0.098Sign 0.925 0.859 0.446 0.084 0.863 0.811 0.423 0.082

�= 1 ADF 0.111 0.073 0.036 0.046 0.111 0.073 0.036 0.046Wilcoxon 0.984 0.969 0.698 0.097 0.960 0.943 0.661 0.095Normal 0.974 0.955 0.660 0.100 0.938 0.921 0.623 0.102Sign 0.983 0.971 0.700 0.085 0.935 0.943 0.668 0.089

�= 0:8 ADF 0.107 0.084 0.043 0.050 0.107 0.084 0.043 0.050Wilcoxon 0.991 0.993 0.909 0.094 0.974 0.979 0.887 0.090Normal 0.989 0.991 0.890 0.103 0.971 0.975 0.859 0.103Sign 0.990 0.991 0.928 0.086 0.915 0.973 0.906 0.086

�= 0:5 ADF 0.111 0.101 0.051 0.049 0.111 0.101 0.051 0.049Wilcoxon 0.976 0.975 0.988 0.091 0.967 0.964 0.976 0.092Normal 0.976 0.978 0.990 0.101 0.971 0.971 0.980 0.102Sign 0.970 0.967 0.980 0.088 0.781 0.933 0.957 0.090

aSample size: 100, number of replications: 5000, nominal signi$cance of the test: 5%, nullmodel: Nyt = + + �1Nyt−1 + et , where et ∼ i.i.d. symmetric stable law.

Although we do not present here any theoretical results on the optimalityof the tests, we may nonetheless compare the power of these rank tests withthat of ADF test. Because of the complex limiting distribution of ADF, wecompute the 5% quantile of the ADF test statistic based on 5000 replicationsof sample size 100. These are reported in Table 6 as the 5% critical value ofthe ADF tests. Clearly, for � = 2, the critical value corresponds to Dickey–Fuller table for $nite variance case. However, the critical value increaseswith the decreasing value of �, indicating the perilous eVect of thicker taildistributions.With these critical values, we computed the empirical power of ADF tests

from the same seed as we compute size and power of the rank tests. Thus,the ADF results in Table 4 are directly comparable with rank tests. It is clearthat ADF tests have extremely low power for all values of �¡ 2. The strikingfeature of Table 4 is not only the enormous gain in power of rank tests over


Table 5Finite sample correction: Zhat version (ST )a

Index of Test statistic l= 2 l= 6stable distribution

Power Size Power Size0.90 0.95 0.99 1.00 0.90 0.95 0.99 1.00

�= 2 Wilcoxon 0.056 0.045 0.032 0.034 0.039 0.031 0.026 0.027(normal) Normal 0.062 0.049 0.036 0.040 0.044 0.034 0.028 0.032

Sign 0.045 0.046 0.044 0.045 0.037 0.038 0.038 0.036

�= 1:2 Wilcoxon 0.874 0.807 0.387 0.059 0.802 0.741 0.347 0.051Normal 0.807 0.746 0.342 0.053 0.725 0.667 0.296 0.043Sign 0.883 0.816 0.406 0.062 0.789 0.765 0.373 0.057

�= 1 Wilcoxon 0.976 0.951 0.641 0.063 0.944 0.923 0.594 0.057Normal 0.952 0.923 0.585 0.056 0.900 0.876 0.530 0.048Sign 0.981 0.960 0.671 0.071 0.896 0.934 0.631 0.068



aSame con$guration as in Table 4.

Table 6Critical values of ADF test statistic

�= 2 �= 1:2 �= 1 �= 0:8 �= 0:5

−2:924662 −3:196315 −3:324561 −3:522408 −4:254601

ADF test, but also the fact that rank tests can spectacularly discriminate evenin the near unit root case for values of �¡ 2.

5. Conclusion

In the absence of a satisfactory formal statistical procedure to distinguishbetween $nite and in$nite variance process, the choice of the modeling ap-proach is conventionally left to the taste and philosophical conviction of anindividual researcher. However, for unit root case, the feasible version (ST )for the in$nite variance case is quite similar to the test statistics, ST forthe $nite variance case, with ST having some advantage over ST in termsof size for �¡ 2. In a situation when a researcher has a strong prior or


sample evidence towards one case over the other, he may appropriately choosebetween ST and ST . As a matter of fact, with a little premium in terms ofslightly inYated size the ST may serve as powerful test statistics in detect-ing unit roots problem without having to worry about the $nite and in$nitevariance cases. The proposed test statistics do not require the estimation of� parameter. The asymptotic distribution of the test statistics are Gaussian.They are shown to have reliable size and have much improved power thanthe conventional ADF test statistic for a wide variety of stable distributionsthus extending the applicability of the tests proposed in HK (1997) to thedomain of in$nite variance errors.

Acknowledgements

The author would like to express his appreciation to Roger W. Koenker,Stephen Portnoy, and Paul Newbold for stimulating discussions regardingvarious aspects of this work. This research has been partially supported byURG 95-96 grant from Illinois State University.

References

Adichie, J.N., 1986. Rank tests in linear models. In: Krishnaiah, P.R., Sen, P.K. (Eds.),Handbook of Statistics, Vol. 4. Elsevier, New York.

Campbell, B., Dufour, J.-M., 1995. Exact nonparametric orthogonality and random walk tests.Review of Economics and Statistics 77, 1–16.

Campbell, B., Dufour, J.-M., 1997. Exact nonparametric tests of orthogonality and random walkin the presence of a drift parameter. International Economic Review 38, 151–173.

Chambers, J.M., Mallows, C.L., Stuck, B.W., 1976. A method for simulating stable randomvariables. Journal of the American Statistical Association 71, 340–344.

Chan, N.H., Tran, T., 1989. On the $rst order autoregressive process with in$nite variance.Econometric Theory 5, 354–362.

ChernoV, H., Savage, I.R., 1958. Asymptotic normality and ePciency of certain non-parametrictest statistics. Annals of Mathematical Statistics 29, 972–994.

Davis, R.A., Knight, K., Liu, J., 1992. M-estimation for autoregression with in$nite variance.Stochastic Processes and their Applications 40, 533–558.

Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time serieswith a unit root. Journal of the American Statistical Association 74, 427–431.

Dufour, J.M., 1981. Rank test for serial dependence. Journal of Time Series Analysis 2, 117–128.

DuMouchel, W.H., 1983. Estimating the stable index � in order to measure tail thickness: acritique. The Annals of Statistics 11, 1019–1031.

Gutenbrunner, C., Jure0ckov1a, J., 1992. Regression rank-scores and regression quantiles. Annalsof Statistics 20, 305–330.

Gutenbrunner, C., Jure0ckov1a, J., Koenker, R.W., Portnoy, S., 1993. Tests of linear hypothesesbased on regression rank scores. Journal of Nonparametric Statistics 2, 307–331.

H1ajek, J., 0Sid1ak, Z., 1967. Theory of Rank Tests. Academia, Prague.


Hallin, M., Puri, M.L., 1992. Rank tests for time-series analysis: a survey. In: Brillinger, D.,et al. (Eds.), New Directions in Time Series Analysis, Vol. 45. Springer, New York, pp.111–153.

Hallin, M., Puri, M.L., 1994. Aligned rank tests for linear models with autocorrelated errorterms. Journal of Multivariate Analysis 51, 175–237.

Hasan, M.N., Koenker, R.W., 1997. Robust rank tests of the unit root hypothesis. Econometrica65, 133–161.

Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. TheAnnals of Statistics 3, 1163–1174.

Hodges, J.L., Lehmann, E.L., 1956. The ePciency of some nonparametric competitors of thet-test. Annals of Mathematical Statistics 27, 324–335.

Janicki, A., Weron, A., 1994. Simulation and Chaotic Behavior of �-Stable Processes. MercelDekker, New York.

KlGuppelberg, C., Mikosch, T., 1996. The integrated periodogram for stable processes. TheAnnals of Statistics 24, 1855–1879.

Knight, K., 1989. Limit theory for autoregressive parameters in an in$nite variance randomwalk. Canadian Journal of Statistics 17, 261–278.

Knight, K., 1991. Limit theory for M-estimates in an integrated in$nite variance process.Econometric Theory 7, 200–212.

Koenker, R., 1996. Rank tests for linear models. In: Rao, C.R., Maddala, G.S. (Eds.), Handbookof Statistics, Vol. 15. Elsevier, New York.

Koenker, R.W., Bassett, G., 1978. Regression quantiles. Econometrica 46, 33–50.Koenker, R.W., d’Orey, V., 1987. Computing regression quantiles. Applied Statistics 36, 383–

393.Koenker, R.W., d’Orey, V., 1993. Remark on algorithm 229, Computing dual regression

quantiles and regression rank scores. Applied Statistics 42, 428–432.Koul, H., Saleh, A.K.M.E., 1995. Autoregression quantiles and related rank scores processes.

Annals of Statistics 23, 670–689.Martin, R.D., 1980. Robust estimation of autoregressive models. In: Brillinger, D.R., Tiao,

G.C. (Eds.), Directions in Time Series. Institute of Mathematical Statistics, Hayward, CA,pp. 228–254.

Martin, R.D., 1981. Robust methods for time series. In: Findley, D.F. (Ed.), Applied TimeSeries II. Academic Press, New York, pp. 683–759.

McCulloch, J.H., 1997. Measuring tail thickness to estimate the stable index �: a critique.Journal of Business and Economic Statistics 15, 74–81.

Mikosch, T., Gadrich, T., KlGuppelberg, C., Adler, R.J., 1995. Parameter estimation for ARMAmodels with in$nite variance innovations. The Annals of Statistics 23, 305–326.

Phillips, P.C.B., 1987. Time series regression with a unit root. Econometrica 55, 277–301.Phillips, P.C.B., 1990. Time series regression with a unit root and in$nite-variance errors.

Econometric Theory 6, 44–62.Puri, M.L., Sen, P.K., 1985. Nonparametric Methods in General Linear Models. Wiley, New

York.Resnick, S., Greenwood, P., 1979. A bivariate stable characterization and domains of attraction.

Journal of Multivariate Analysis 9, 206–221.

rank tests of unit root hypothesis with infinite variance errors

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